We apply two different approaches to quantify mixing in
a shallow water model of the stratosphere.
These are modified Lagrangian mean (MLM) theory, and a
technique referred to as ``reverse domain filling with
local gradient reversal'' (RDF-LGR). The latter
is similar to a previously existing technique using contour
advection and contour surgery.

It is first proved that in an inviscid shallow water atmosphere
subject to mass sources and sinks, if the mass enclosed by a
potential vorticity (PV) contour is steady in time, then the
integral of the mass source over the area enclosed by the contour
must be zero. Next, the MLM and RDF-LGR approaches are
used to diagnose the time-averaged transport across PV contours in the model
simulations.

The model includes a sixth-order hyperdiffusion
on the vorticity field. Except
in a thin outer ``entrainment zone'', the hyperdiffusion term has only a
very weak effect on the MLM mass budget of the polar vortex.
In the entrainment zone, the
hyperdiffusion term has a significant effect. The RDF-LGR results
capture this behavior, providing good quantitative estimates
of the hyperdiffusion term, which is equivalent to the degree of
radiative disequilibrium at a PV contour. This agreement
shows that the main role of the hyperdiffusion is to
``mop up'' the filaments which are produced by the essentially
inviscid large-scale
dynamics. All calculations are repeated for two values of the
hyperdiffusion coefficient which differ by a factor of 50, with
little difference in the results. This suggests
that the amount of material
entrained from the vortex edge into the surf zone does
not depend on the details of the small-scale dissipation, as long
as it is sufficiently weak and has some degree of scale-selectivity.